GERBES ON QUANTUM GROUPS

arXiv:math/0308235v1 [math.DG] 25 Aug 2003

GERBES ON QUANTUM GROUPS

Jouko Mickelsson Mathematical Physics, KTH, SE-106 91 Stockholm, Sweden August 25, 2003

Abstract We discuss an approach to quantum gerbes over quantum groups in terms of q-deformation of transition functions for a loop group bundle. The case of the quantum group SUq (2) is treated in some detail.

0. Introduction In view of the recent extensive activity in the theory of gerbes and their applications in quantum field theory it is not surprising that the question arises whether there is some sort of object which could be called a ’quantum gerbe’. One should understand ’quantum’ here really meaning a deformation depending on a real parameter q since a gerbe already is in a sense a quantum object: It has a well defined ’quantum number’ given by its Dixmier-Douady class and this class in the physical applications is related to the chiral anomaly in quantum field theory. A proposal for a quantum gerbe was discussed in [ABJS] in terms of deformation quantization. The underlying base space of the gerbe was replaced by noncommutative space defined as a star product algebra. A noncommutative version of a line bundle was defined, [JSW], and then a gerbe was defined as a system of local noncommutative line bundles obeying a certain cocycle condition, imitating the corresponding cocycle condition for the undeformed case. In this paper we follow a different route to quantum gerbes. We start from the alternative definition of a gerbe (without connection and curvature) as a principal P U (H) bundle over a manifold M, where P U (H) is the projective unitary group of a complex Hilbert space H. The gerbe class can be nontorsion only if H is infinite-dimensional. The equivalence classes 1 of such a bundles are parametrized by elements in H 3 (M, Z). The characteristic class is called the Dixmier-Douady class. Typeset by AMS-TEX

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JOUKO MICKELSSON

Concretely, the principal P U (H) bundle can be obtained from a loop group bundle using a projective unitary representation of the loop group. In this paper we shall construct ’quantum loop group bundles’ using transition functions in a group of matrices with entries in a quantum group. The local approach to gerbes consists of an open cover {Uλ } of M and a set of line bundles Lλλ′ over the intersections Uλ ∩ Uλ′ with given isomorphisms (1)

Lλλ′ ⊗ Lλ′ λ′′ = Lλλ′′

on triple intersections. The relation to the global description is as follows. There is a canonical central extension 1 → S 1 → U (H) → P U (H) → 1 of the projective unitary group which gives a canonical complex line bundle L over P U (H). The transition functions φλλ′ : Uλ ∩ Uλ′ → P U (H) of the principal bundle can then be used to pull back the line bundle L over P U (H) to a local line bundle Lλλ′ over Uλ ∩ Uλ′ . The group structure on U (H) defines an identification of Lλλ′ ⊗ Lλ′ λ′′ with Lλλ′′ . In the case of the quantum group SUq (2) we shall see how the the local line bundle approach is related to the loop group bundle construction.

1. The gerbe over SU (n): The undeformed case Let us next consider the case when M = G is a compact connected Lie group. Let P be the space of all smooth paths f in G starting from f (0) = 1 and with an arbitrary endpoint f (1) ∈ G. We also require that f −1 df is a smooth periodic function, that is, it defines a vector potential on the unit circle S 1 . Thus we may identify P as the contractible space A of smooth vector potentials on S 1 with values in the Lie algebra of G. P is the total space of a principal bundle over G with fiber equal to the group ΩG of smooth based loops in G. Based means that f (0) = f (1) = 1. Since P = A is contractible, it is a universal bundle for ΩG. Assume that ψ : ΩG → P U (H) is a projective representation of ΩG in the Hilbert space H. We can then define


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an associated principal P U (H) bundle over G in the usual way, with total space Q = P ×ψ P U (H). This is the way gerbes appear in canonical quantization in field theory, [M, CM1-2, CMM]. We shall define a quantum gerbe over a quantum group SUq (N ) in terms of ’local transition function’ with values in a loop group. This means that the transition functions are loops in a group of matrices with entries in the Hopf algebra SUq (n). To warm up, we start from the classical case by giving explicit formulas for the transition functions. In the case of SU (n) it is sufficient to select n open sets to cover the base. We choose n different points λ1 , . . . λn 6= 1 in the unit circle in the complex plane, ordered counter clockwise, such that the product λ1 λ2 . . . λn 6= 1. For i = 1, 2, . . . n let Ui be the subset of SU (n) consisting of matrices g such that λi is not an eigenvalue of g. This gives an open cover: If g is not in any of the sets Ui then all λi ’s are eigenvalues of g and so det(g) = λ1 λ2 . . . λn 6= 1, a contradiction. An each open set Ui we have a trivialization of the bundle P → SU (n). Let hi (t) be any fixed smooth contraction of the set S 1 \ {λi } with hi (0) the constant map sending S 1 \ {λi } to the point 1 and hi (1) the identity map. Let d = (d1 , . . . , dn ) be a diagonalization of g ∈ Ui , g = AdA−1 . Then we set ψi (g)(t) = A(hi (t)(d1 ), . . . , hi (t)(dn ))A−1 and this defines a path in Ui joining g to the neutral element, i.e., we have a local section ψi : Ui → P. The transition function on Ui ∩ Uj is fixed by ψj (g) = ψi (g)φij (g) and it takes values in ΩG. Remark 1 Allowing the paths lie in the bigger group GL(n, C) leads to a technical simplification in the construction of transition functions. First, for each index i we have the contraction of Ui to the point −λi · 1 given by ψi (g)(t) = −(1 − t)λi · 1 + tg. Note that the matrix ψi (g)(t) is really invertible for each g ∈ Ui and 0 ≤ t ≤ 1 by the spectral property Spec(g) ⊂ S 1 . Since GL(n, C) is connected, we can deform the constant map ψi (g)(0) = −λi · 1 to the constant map g 7→ 1 in an obvious way. Putting these together we obtain a homotopy connecting the constant map g 7→ 1

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JOUKO MICKELSSON

to the indentity map g 7→ g in Ui . The transition functions are defined as before, φij (g) = ψi (g)−1 ψj (g), now with values in the loop group of GL(n, C). A third way to define the trivializations and transition functions is to use functional operator calculus. If λ ∈ S 1 is not in the spectrum of a unitary matrix g then the logarithm of g can be defined as Z log(z) dz, (2) log(g) = γ g−z where γ is a closed loop in the complex plane C \ {0} which encircles every point in S 1 \ {λ} with winding number one; the loop crosses the point λ twice such that the ray R from the origin to λ is tangential to γ at λ but otherwise γ does not cross the ray; then we can define the branch of log(z) as an analytic function with a cut along R and (g − z)−1 is nonsingular along γ. Denoting X = log(g) we have a smooth path ψi (g)(t) = etX connecting 1 to the point g ∈ Ui for λ = λi . As mentioned in the beginning, the local line bundle construction of a gerbe over SU (n) is obtained by pulling back the (level k) central extension of the loop group by the transition functions. However, there is simpler construction which leads to the same gerbe. As shown in [CM1] the basic gerbe (level k = 1) is constructed using a family of Dirac operators parametrized by points on G. For G = SU (n) the Dirac operator Dg attached to g ∈ G has domain consisting of smooth functions on the interval [0, 1] with values in Cn and boundary conditions ψ(1) = gψ(0). The Dirac operator d on the interval. If g = 1 the spectrum is simply the differentiation Dg = −i dx

has multiplicity n and consists of the numbers 2πm, m ∈ Z. For arbitrary g the spectrum consists of 2πm + µi where µ1 , . . . , µn are the eigenvalues of −i log(g). ′

If now λ, λ′ are two distinct points on the unit circle, λ = eiµ , λ′ = eiµ , with 0 ≤ µ, µ′ < 2π, then the eigenvalues of the Dirac operator Dg with g ∈ Uλ ∩ Uλ′ in the spectral interval ]µ, µ′ [ match the eigenvalues of g in the segment ]λ, λ′ [ of the unit circle. It follows that the determinant line DETµµ′ for the Dirac operators Dg is naturally isomorphic to the top exterior power Λtop (Eλλ′ ) of the corresponding spectral subspace for g. Combining with [CM1] we get the equivalences: Theorem. The basic gerbe over G = SU (n), corresponding to the Dixmier-Douady class given by the generator of H 3 (SU (n), Z), can be given in three equivalent ways: (1) as the P U (H) bundle over G obtained as an associated bundle to the univer-


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sal bundle A → G through the basic representation of the affine Kac-Moody group based on G (2) as the local system of complex line bundles defined as above by Dirac operators on the unit interval, parametrized by boundary conditions g ∈ G (3) as the system of local line bundles formed from top exterior powers of the spectral subspaces of g ∈ G corresponding to open segments of the unit circle. Let us consider the case G = SU (2) in more detail. It is sufficient to choose a cover consisting of two open sets U± consisting of points g 6= ±1. The overlap is homotopic to the equator S 2 in SU (2) = S 3 . The basic transition function for a loop group bundle on SU (2) is then (3)

φ(x)(t) = cos(2πt) + ix sin(2πt) for 0 ≤ t ≤

1 2

x3 x1 + ix2 and φ(x)(t) = diag(e ,e ) for ≤ t ≤ 1. Here x = . x1 − ix2 −x3 Actually, to make the function smooth at t = 1/2 and the end points one should 2πit

−2πit

1 2

replace the variable t by f (t) where f is any smooth monotonous function on the interval [0, 1] such that f (0) = 0, f (1) = 1, and all the derivatives vanish at the points t = 0, 1/2, 1.

2. The gerbe over the quantum group SUq (n) We assume 0 < q < ∞ and denote by SUq (n) the standard Hopf algebra of the quantum special unitary group. It is given in terms of generators and relations as follows. The generators are elements gij indexed by i, j = 1, 2, . . . n with defining relations

gim gik = qgik gim , gjm gim = qgim gjm (4)

gim gjk = gjk gim , gik gjm − gjm gik = (q −1 − q)gim gjk

for i < j and k < m. In addition, the quantum determinant (5)

detq =

X

σ∈Sn

(−q)−ℓ(σ) g1σ(1)g2σ(2) . . . gnσ(n)

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JOUKO MICKELSSON

is identified as the unit element in the algebra. The counit is defined by ǫ(gij ) = δij and the antipode is defined as S(gij ) = (−1)i+j Xji ,

(6)

where Xij is the (n − 1) × (n − 1) quantum minor of the matrix (gij ), i.e., the quantum determinant of the submatrix obtained by deleting the i:th row and j :the column from (gij ). The star algebra structure is given as ∗ gij = S(gji ).

(7)

In particular, by the definition of an antipode, P ′ ′′ (x) x ⊗ x , and so

P

(x)

x′ S(x′′ ) = ǫ(x)·1 where ∆(x) =

∗ gij gik = δjk · 1.

(8)

The coproduct ∆ is defined by matrix multiplication, ∆(gij ) =

X

gik ⊗ gkj .

k

For more details, see [KS]. In the case n = 2 the standard notation is a = g11 , b = g12 , c = g21 , d = g22 with b∗ = −qc, a∗ = d. The determinant condition is ad − qbc = 1 and the commutation relations are (9)

ab = qba, ac = qca, bd = qdb, cd = qdc, bc = cb, ad − da = (q − q −1 )bc.

We shall now define a ’quantum gerbe’ over the quantum group SUq (2) as a quantum projective bundle over SUq (2). This is achieved by giving the transition function on the ’equator’ of SUq (2). The quantum equator is defined as a quotient algebra of SUq (2), [HMS]. Let I ⊂ SUq (2) be the 2-sided ideal generated by the single element b−b∗ = b+qc. Then one can show that the standard quantum sphere Sq2 is isomorphic to SUq (2)/I, [HMS]. Explicitly, the generators of Sq2 are K = K ∗ and L with the defining relations (10)

LL∗ + q 2 K 2 = 1, L∗ L + K 2 = 1, LK = qKL.


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One can check that K = c + I, L = a + I in SUq (2)/I indeed satisfy the relations above. In the classical case, q = 1, going to the quotient means that we set one of the coordinates, namely Im(b), equal to zero reducing the 3-sphere to the equatorial 2-sphere. The transition function is defined as in the undeformed case: Define the quantum 2 × 2 matrix (11)

x=

qK L∗

L −K

.

It is easy to check from the defining relations that x∗ = x (combination of matrix transposition and star operation on Sq2 ) and that x2 = 1. It follows that cos(2πt) + ix sin(2πt) (0 ≤ t ≤ 1/2) is unitary and in combination with the path t 7→ diag(e2πit, e−2πit ) (1/2 ≤ t ≤ 1) defines a loop in unitary 2 × 2 matrices with coefficients in the algebra Sq2 . In the classical undeformed case the transition function can from be extended 0 i the equator to the open set U+− = SU (2) \ {±σ} where σ = . First one i 0 extends the function x to U+− by writing (12)

x=f

−1

1 2

(b + b∗ ) a

a∗ − 21 (b + b∗ )

,

where f = [1 + 12 (b − b∗ )2 ]1/2 . This function is singular at the points where Im(b) = ±i, that is, at opposite poles of S 3 = SU (2). It satisfies x∗ = x =x2 and its b d restriction to the equator defined by Im(b) = 0 is equal to . We may a −b also view x as a matrix with entries in an algebraic extension of the commutative algebra of functions on SU (2) by a single element f satisfying the defining relation f 2 = 1 + 14 (b − b∗ )2 . That is, we add certain functions with singularities at the poles to the algebra generated by the elements a, b, c, d. The generalization of the above formula to the q-deformed case is given as (13)

x=

1 2q (b

+ b∗ )fq−1 f −1 a

a∗ f −1 − 21 (b + b∗ )f −1

where f is same as before and fq = [1 + 4q12 (b −b∗ )2 ]1/2 . This means that the matrix x has entries in an algebraic extension of SUq (2) by the elements f, fq satisfying the relations f 2 = 1 + 41 (b − b∗ )2 and likewise for fq .

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JOUKO MICKELSSON

As in the undeformed case, we can think of the quantum gerbe coming from a K-theory class defined by a family of Dirac type operators. The family of operators d is again given by the differentiation D = −i dt but now acting on 2-component

spinors with coefficients in the quantum group SUq (2). The family of boundary conditions is ψ(1) = gψ(0) a b where g is the quantum matrix g = . So in a sense the K-theory class in c d K 1 (SUq (2)) is tautological: It is the unitary 2×2 matrix g given by the ’coordinates’ of the quantum group. Note that the K-theory of SUq (2) when q > 0 is the same as for q = 1, K 1 (SUq (2)) = Z = K 0 (SUq (2)), [MNW2]. The quantum line bundle on the equator Sq2 is defined as the projection P = 1 (1 2

+ x) with coefficients in Sq2 . The property P 2 = P follows from x2 = 1. Of

course the case of SUq (2) is very simple since there is only one ’overlap’ Sq2 and we do not need to bother about the generalization of (1) to the quantum group case. Remark 2 The rank one projectors in the q deformed case define only right Sq2 modules and not bimodules since the left multiplication by the elements in the algebra Sq2 does not commute with the multiplication by P on Sq2 ⊕ Sq2 . For this reason the tensor products of line bundles over Sq2 are not canonically defined. Since there is no underlying smooth manifold, the Dixmier-Douady class cannot be defined in terms of de Rham forms as in the undeformed case. Instead, one can use a cyclic cocycle c3 to compute the quantum invariant of the gerbe by pairing c3 with the K-theory class of the unitary matrix g. This was in fact already done in [Co] (in the case 0 < q < 1) and I will not repeat (the rather complicated) calculations here. Next we want to generalize the construction to G = SUq (n). This is achieved by a generalization of the method in Remark 1. Let λ ∈ S 1 . Define an algebraic extension SUq (n)λ of SUq (n) by requiring that the equation (g − λ)h = 1 has a solution h as a n × n matrix with entries in the extended algebra. In the classical case this means that we allow singularities for det(g −λ), i.e., we restrict the domain of functions on SU (n) to the open subset specified by λ ∈ / Spec(g). Actually, we need to consider the algebra SUq (n) as a ∗-algebra completion of the algebra defined by the relations (4). Then g − tλ has an inverse for any real


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number t 6= 1 since g is a unitary matrix. We have then a homotopy (14)

ψt (g, λ) = −(1 − t)λ + tg

connecting ψ0 (g, λ) = −λ to the identity map g 7→ g = ψ1 (g, λ). The homotopy is defined in the space of nonsingular matrices with entries in SUq (n)λ . We extend this homotopy to a path connecting the constant map g 7→ 1 to the identity g 7→ g in an obvious way, along the path ψt = (−λ)t , 0 ≤ t ≤ 1 connecting the neutral element to λ. The ’transition functions’ are defined as in the undeformed case, φλλ′ (g)(t) = ψt (g, λ)ψt(g, λ′ )−1 . The transition functions are loops of n × n matrices with coefficients in the extension SUq (n)λλ′ defined by the inverses (g − λ)−1 , (g − λ′ )−1 . Acknowledgements This work was supported by Erwin Schr¨odinger Institute for Mathematical Physics and CPT in Luminy (Universite de Provence) and I want to thank especially Thomas Sch¨ ucker for hospitality in Luminy.

References [ABJS] P. Aschieri, I. Bakovic, B. Jurco, P. Schup: Noncommutative gerbes and deformation quantization. hep-th/0206101 [CM2] A. L. Carey and J. Mickelsson: The universal gerbe, Dixmier-Douady class, and gauge theory. Lett.Math.Phys. 59, 47-60 (2002). hep-th/0107207 [CM1] A. L. Carey and J. Mickelsson: A gerbe obstruction to quantization of fermions on odd dimensional manifolds with boundary. Lett.Math.Phys.51, 145160,(2000). hep-th/9912003 [CMM] A.L. Carey, J. Mickelsson, and M. Murray: Bundle gerbes applied to field theory. Rev.Math.Phys.12, 65-90 (2000). hep-th/9711133 [Co] A. Connes: Cyclic Cohomology, Quantum group Symmetries and the Local Index Formula for SUq (2). math.QA/0209142 [HMS] P.M. Hajac, R. Matthes, W. Szymanski: Graph C*-algebras and Z/2Zquotients of quantum spheres. math.QA/0209268 [JSW] B. Jurco, P. Schupp, J. Wess: Noncommutative line bundles and Morita equivalence. Lett.Math.Phys.61, 171-186 (2002). hep-th/0106110

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[KS] L.I. Korogodski and Ya. S. Soibelman: Algebras of functions on quantum groups. Part I. Mathematical Surveys and Monographs, 56. American Mathematical Society, Providence, RI, 1998. [MNW1] T. Masuda, Y. Nakagami, and J. Watanabe: Noncommutative differential geometry on the quantum two sphere of Podles. I: An algebraic viewpoint. K-Theory 5, 151-175 (1991) [MNW2] T. Masuda, Y. Nakagami, and J. Watanabe: Noncommutative differential geometry on the quantum SU (2). I: An algebraic viewpoint. K-Theory 4, 157-180 (1990) [Mi] J. Mickelsson: On the Hamiltonian approach to commutator anomalies in (3 + 1) dimensions. Phys.Lett. B241, 70-76, (1990)